The mathematical investigation of cancer began in the 1950s when several investigators set out to explain the age-dependent incidence curves of human cancers. Early milestones were Armitage and Doll's observation that log-log plots of cancer incidence curves were linear, and Knudson's statistical study of retinoblastoma which led to the two-hit hypothesis of tumor suppressor gene inactivation. Subsequent theoretical and experimental work has proven the hypothesis that for most cancer types, the accumulation of several mutations is necessary not only for cancer initiation, progression, and metastatic spread, but also for the emergence of resistance against chemotherapeutics. There is a large and growing mathematical literature on cancer models, but most of the analyses are performed for Markovian models of a homogeneously mixing population of a constant size or for an exponentially growing branching process. The main aim of this proposal is to develop flexible mathematical analyses to understand the changes in predicitons when the cell population is divided into different types of cells, when residence times in the different compartments of the model are not exponential, or when the spatial structure of growing tumors is taken into account. We will also consider the implications of these considerations for tumor heterogeneity, and for the types of cells that are the most likely targets of mutations that initiate disease or treatment failure.